Recursive Preferences and Balanced Growth1

Roger E. A. Farmer2

UCLA, and CEPR

rfarmer@econ.ucla.edu

Amartya Lahiri

UCLA

lahiri@econ.ucla.edu

This Draft: May 2003

1Both authors acknowledge the support of UCLA academic senate grants. We thank Mick

Devereux, Martin Schneider and Ping Wang for helpful feedback.2Corresponding Address: Roger E. A. Farmer, Department of Economics, 8283 Bunche Hall,

UCLA, Los Angeles, CA 90095-1477; Telephone: (310) 825-6547; Fax: (310) 825-9528.

Abstract

We study a class of utility functions that are defined recursively by an aggregator W (x, y)

where ut = W (ct, ut+1). In single-agent economies it is known that a sufficient condition

for the existence of a balanced growth path is that utility should be homogenous of degree

γ. In the context of a multi-agent economy we show that this restriction implies that either

a balanced growth equilibrium fails to exist or all agents have the same constant discount

factor. We suggest a generalization of recursive preferences wherein the intertemporal utility

function is time dependent. Within this class we establish that there may exist a balanced

growth equilibrium even if agents are different. We give an example of our approach in the

international context in which time dependence occurs because countries care about their

relative position in the world income distribution.

1 Introduction

Two central features of most modern macroeconomic models are (a) the assumption of infi-

nitely lived families with intertemporally separable utility functions; and (b) an environment

that allows for balanced growth. While the first stems primarily from a desire for simplicity,

the balanced growth construct originates in the seminal work of Kaldor [4] who stressed that

balanced growth provides a good characterization of the long run development experience of

the currently industrialized countries. According to the Kaldor growth facts, development

paths are characterized by the constancy of growth rates, factor income shares and capital-

output ratios. From a modelling standpoint, balanced growth is an attractive feature since

it makes otherwise complicated environments simple to analyze.

It is well known that the assumption of time separability of preferences is restrictive since

all agents must have the same rate of time preference if a model in this class is to display

a non-degenerate wealth distribution. If rates of time preference are different across agents

then the most patient family will asymptotically own all of world wealth. A large literature

starting with Koopmans [6], Uzawa [10] and more recently Epstein and Hynes [3], and Lucas

and Stokey [7] has generalized the choice over intertemporal consumption sequences to the

case of recursive preferences in which the rate of time preference is a function of the agent’s

consumption sequence. Lucas-Stokey [7] and Epstein-Hynes [3] have used these preferences

to construct examples of economies for which there exists a non-degenerate asymptotic wealth

distribution.

Lucas and Stokey [7] studied an environment in which agents have recursive preferences

defined over bounded consumption sequences. Within this environment they proved the ex-

istence of a non-degenerate income distribution; but their result does not apply to economies

with growth. Boyd [1] generalized recursive utility to a class of preferences defined over un-

1

bounded consumption sequences and his work does permit the study of environments that

permit growth. Using Boyd’s results, Dolmas [2] proved that there exists a balanced growth

path in the representative agent growth model if there is a recursive representation of pref-

erences, ut = W (ct, ut+1) for which the aggregator W (x, y) is homogeneous of the form

W (λx,λγy) = λγW (x, y).

In this paper we study the conditions under which recursive preferences, balanced growth,

and heterogenous preferences (i.e., multiple agents) can coexist. When preferences are

described by homothetic utility functions, (a necessary condition for the existence of balanced

growth) we show that either the asymptotic wealth distribution in a multi-agent economy

is degenerate or all agents have the same constant discount rate. This result is troubling

given that a key motivation for introducing recursive preferences is to permit heterogeneity

without inducing a degenerate wealth distribution.

To recover the property of non-degeneracy of the wealth distribution we introduce an

exogenous time-dependence into the utility function and show that this formulation permits

the coexistence of recursive preferences, preference heterogeneity, and balanced growth. We

suggest an application of time dependence in which agents care about their relative position

in the world wealth distribution; in this context world wealth enters utility as an external

effect. Using this example, we show that our formulation is consistent with a determinate

distribution of world wealth in the context of both exogenous and endogenous growth models.

The paper is structured as follows. Sections 2 and 3 describe basic assumptions on

aggregator functions and technology sets. These sections are relatively technical and could

be skipped by a reader who has some familiarity with the idea of recursive preferences and

who is interested in the main economic ideas. Section 4 provides necessary and sufficient

conditions for the consistency of general non-separable utility functions with balanced growth

paths. This section also contains the intuition behind the main theorem of the paper, proved

2

in Section 5, that balanced growth requires all agents to have the same constant rate of time

preference. Sections 6 and 7 provide a generalization of recursive preferences in which we

suggest a non autonomous formulation of the utility function and show that this formulation

is consistent with balanced growth. Finally, in Section 8 we provide two examples of our

approach in which time enters the utility function because agents care about their relative

position in the world income distribution. In these examples, the discount factor of an agent

becomes lower as he becomes wealthy relative to other agents in the world economy. Section

9 provides a few concluding comments.

2 Preliminaries

Our approach follows Lucas and Stokey [7] as adapted by Boyd [1]. Lucas and Stokey

show how to define a utility function recursively from an aggregator function W (x, y) that

is bounded and satisfies certain additional properties. Following their approach, consider the

following candidate space of admissible consumption sequencesC = {C ∈ X | X ⊂ l∞, xt ≥ 0}where l∞ is the space of bounded sequences with the norm kxk = sup |xt|.1 Let B be the

space of continuous bounded functions u : X→ R with the norm

kuk = supx∈X

|u (x)| ,

and define an operator TW : B→ B as

(TWu) (C) = W (πC, u (SC)) . (1)

In equation (1) π is the projection operator πX = x1, and S is the shift operator SX =

(x2, x3, ...) . Lucas and Stokey show that under their assumptions about the properties ofW,1We use the convention that uppercase letters represent vectors in R∞, lowercase letters are scalars and

boldface letters represent sets.

3

TW is a contraction and hence the aggregator W can be used to construct a unique utility

function u (C) .

For our purposes the approach in Lucas and Stokey is restrictive since we wish to study

balanced growth. We need to allow both consumption sequences and aggregator functions

W (x, y) to be unbounded. Boyd [1] shows how to apply a modified version of the contraction

mapping theorem, the weighted contraction mapping theorem, that applies to a much larger

class of aggregators than those permitted by Lucas and Stokey. Boyd allows the consumption

set to be larger than l∞ but smaller than R∞ so that there is an appropriate notion of

convergence of sequences and convergence of continuous functions defined on this larger

space. The appropriate consumption set, found by bounding the rate of growth of feasible

consumption sequences, is

X (g) =nX ∈ R∞ | |X|g <∞

owhere |X|g = sup |xt/gt−1| for g ≥ 1 is the g-weighted l∞ norm. Boyd proposes the followingmetric on this space.

Definition 1 (Boyd) Let f ∈ c (A,B) , where c (A,B) is the space of continuous functionsfrom A to B. Suppose ϕ ∈ c (A,B) with B ⊂ R and ϕ > 0. The function f is ϕ-bounded

if the ϕ-norm of f, kfkϕ = sup {|f (x)| /ϕ (x)} is finite. The ϕ-norm turns cϕ (A,B) =

{f ∈ c(A,B) :f is ϕ-bounded} into a complete metric space.

In growing economies with a constant returns-to-scale neoclassical technology one can

pick g to equal the growth factor of technological progress.

3 Assumptions on Technology and Preferences

In this section we describe a set of assumptions on preferences and technologies that are

sufficient to guarantee the existence of an equilibrium in a representative agent economy.

4

We assume that the net output of the economy is characterized by a function f (kt, gt) where

g > 1 represents the growth factor of technological progress. At date t we assume that

output is divided between consumption ct and next period’s capital stock kt+1

ct + kt+1 = f¡kt, g

t¢, t = 1, 2...

3.1 Technology

We assume the following properties of the production function f :

Assumption P1 f is continuous

Assumption P2 f is strictly concave

Assumption P3 f is twice continuously differentiable

Assumption P4 f is linearly homogeneous

Assumption P5 limk→∞ fk (k, a) = 0

Assumption P6 limk→0 fk (k, a) =∞

Remark 1 Assumptions P1 − P4 are standard properties of neoclassical production func-tions. Assumptions P5 and P6 (the Inada conditions) ensure an interior solution.

We now turn our attention to finding a suitable upper bound on the rate at which

consumption sequences are permitted to grow. Define the production correspondence P (k)

to be the set of vectors K ∈ R∞ that are feasible given an initial capital stock k.

P (k) =©K ∈ K | 0 ≤ kt+1 ≤ f

¡kt, g

t¢, x1 = k

ªand let

F (k) =©C ∈ C | 0 ≤ ct ≤ f

¡kt, g

t¢− kt+1ª

be the set of feasible consumption paths. Now define f t (k) inductively by the initial condition

f 1 = f (k, 1) and the recursion f t (k) = f (k, gt) ◦ f t−1 (k) . The path of pure accumulation,

5

defined as {f t (k)}∞t=1 , is the sequence of capital stocks that would be attained if all ofsociety’s resources were invested in every period.

For the class of constant returns to scale neoclassical production functions that satisfy

the Inada conditions P5 and P6, f t (k) converges to gtf¡k, 1¢where k is defined by the

equation k = f¡k, 1¢/g. In this case

limt→∞

·f t (k)

gt= k <∞

¸,

and there is no loss to restricting feasible consumption and capital sequences to lie in X (g) ,

the set of sequences in R∞ for which xt/gt is bounded.

3.2 Preferences

We define a class of aggregators W : X(g)×Y → Y that satisfy the following properties.

Assumption U1 W (0, 0) = 0

Assumption U2 W is continuous and increasing in both arguments

Assumption U3 0 ≤ W (x, 0) ≤ A (1 + xη) , A, η > 0

Assumption U4 W is continuous and ϕ-bounded for ϕ (X) = 1 + |X|ηgAssumption U5 |W (x, y)−W (x, y0)| ≤ δ |y − y0| for all x ∈ X(g), y, y0 ∈ Y where δgη < 1

Assumption U6¡TNW y

¢(X) is concave in X for all N and all constants y ∈ Y.

Assumptions U2 and U5 correspond toW1 andW2 in Boyd ([1] page 330). Assumption

U1 restricts us to aggregators that are bounded below and U3 allows us to define a natural

concept of distance using the definition of ϕ-boundedness. Specifically, this assumption

allows us to define a function ϕ:

ϕ (X) = 1 + |X|ηg

where |X|g is the g−weighted l∞ norm.

6

Given assumptions U1 − U5 it follows from Boyd’s Continuous Existence Theorem ([1]

page 333) that there exists a unique u ∈ cϕ such that W (πX, u (SX)) = u (X). Moreover,

from Boyd’s Lemma 1 ([1] page 331), assumption U6 guarantees the concavity of u.

4 Planning Optima and Balanced Growth

In this section we solve a planning problem and ask: What restrictions must we place on

preferences and technology for the solution to this problem to be consistent with balanced

growth?

Consider a representative agent economy in which the agent’s preferences are represented

by an aggregator W that satisfies U1 − U6. A planning optimum for this economy solves

the following problem.

Problem 1

max u (C) = W (πC, u (SC)) (2)

subject to:

ct + kt+1 = f¡kt, g

t¢, t = 1, 2... (3)

k0 = k. (4)

The solution to this problem is characterized by the following set of necessary and suffi-

cient conditions;

Wc (ct, ut+1)−Wu (ct, ut+1)Wc (ct+1, ut+2) fK¡kt+1, g

t+1¢= 0, t = 1, 2.. (5)

limT→∞

uT (C) kT = 0, (6)

where uT (C) ≡Wu (c1, u (SC))Wu

¡c2u

¡S2C

¢¢...Wu

¡cT−1, u

¡ST−1C

¢¢Wc

¡cT , u

¡STC

¢¢. (7)

7

We next turn to the definition of balanced growth.

Definition 2 A balanced growth path is a set of sequences {K,C, Y } for which there existsa triple {k, c, y} such that if k0 = k then

kt = kgt, ct = cg

t, yt = f¡kt, g

t¢= ygt.

Balanced growth is an important assumption because it is a relatively accurate charac-

terization of most industrialized economies. A fair amount is known about technologies

and preferences that are consistent with balanced growth for the case in which preferences

over intertemporal consumption sequences are time separable. Swan [9] proved that the

existence of a balanced growth path implies that exogenous technical progress must be la-

bor augmenting while King, Plosser and Rebelo [5] showed, for the case of additively time

separable preferences, that the utility function must be homogeneous of some degree γ in

consumption. The following lemma extends the results of King-Plosser-Rebelo to the case

of a recursive utility function.

Lemma 1 For a utility function u (C) to display a constant marginal rate of substitution

along a balanced growth path it is necessary and sufficient that the function can be written

as a monotonically increasing function of a linearly homogeneous function. In this case we

say that the function u (C) is homothetic.

Remark 2 Homotheticity of preferences is often defined in terms of preference orderings.

A preference ordering is homothetic if Ca v Cb implies λCa v λCb for all λ > 0.

Proof. Suppose u(C) is homothetic. By our definition of homotheticity, u is a monotone

increasing transformation of a linearly homogenous function. Hence, let

u (C) = F [v(C)] , Fv > 0

where v(C) is linearly homogenous in the consumption sequence C. The marginal rate of

8

substitution is defined by the expression

β (C) =∂u/∂c2∂u/∂c1

=u2 (C)

u1 (C).

Since u (C) is homothetic, u2(C)u1(C)

= v2(C)v1(C)

, and since v (C) is linearly homogeneous, v1(C) and

v2(C) are each homogenous of degree zero in C. Hence,

β (λC) =u2 (λC)

u1 (λC)=v2 (λC)

v1 (λC)=v2 (C)

v1 (C)= β (C) .

Along a balanced growth path SC = gC where S is the shift operator and g is the growth

factor. Hence zero degree homogeneity implies β (SC) = β (gC) = β (C) which establishes

that homotheticity of u (C) is sufficient for the marginal rate of substitution to be constant

along a balanced growth path.

To establish necessity, suppose that the marginal rate of substitution is time invariant,

then

β (C) = β (SC) = β (gC) .

Hence consistency with balanced growth implies that the marginal rate of substitution is ho-

mogeneous of degree 0. Since utility is invariant to monotonically increasing transformations

we consider the class of functions u (C) = F (v (C)) where Fv (v) : R → R is monotonically

increasing and v (C) is quasiconcave in C. Then writing the marginal rate of substitution in

terms of the partial derivatives of u (C) leads to the expression,

β (λC) =F 0 (v (λC)) v2 (λC)F 0 (v (λC)) v1 (λC)

.

Suppose that v1 (C) and v2 (C) are not homogeneous of degree 0 in C. Then either (1)

β (λC) 6= β (C) which is a contradiction or (2) v2 (λC) and v1 (λC) have a common factor-

ization of the form v2 (λC) = f (λC)w2 (λC) , v1 (λC) = f (λC)w1 (λC) where w2 (λC) and

w1 (λC) are each homogenous of degree 0. But in this case we can let w = v through an

appropriate choice of F . Hence u (C) is homothetic.

9

Lemma 1 establishes that prefernces will be homothetic (the marginal rate of substitu-

tion will be constant along a balanced growth path) if and only if the utility function is

homothetic. Lemma 1 has strong implications. Homotheticity means that the rate of time

preference is independent of consumption along any consumption sequence that grows at

a constant rate. This property implies not only that the marginal rate of substitution is

constant along a balanced growth but it is equal to the same constant along any two paths

that grow at the same rate g. In other words, the rate of time preference along balanced

growth paths is a primitive property of preferences. An agent with these preferences, placed

in a small open economy, would not in general have a balanced consumption path unless the

world interest factor happened to equal his discount factor.

Homotheticity is restrictive but it is necessary if we are to construct models in which

the equilibria are consistent with balanced growth. A subclass of homothetic functions is

the class of time-separable homogenous utility functions studied by King-Plosser-Rebelo [5].

A larger subclass is that of homogenous functions that are not necessarily separable. The

following theorem, due to Dolmas, shows how to construct such functions using a class of

recursive aggregators that display a certain homogeneity property. It also establishes the

connection between aggregators of this class and the marginal rate of substitution along

balanced growth paths.

Theorem 1 (Dolmas) SupposeW satisfies assumptions U1−U5 and is such thatW (λx,λγy) =λγW (x, y) for all x and y and all λ > 0, for some γ. Then, the recursive utility function u

exists and is homogeneous of degree γ. If W is also once-differentiable, the marginal rate of

substitution exists and is a constant along a balanced growth path .

Homogenous aggregators of this type have some useful properties that were noted by

Dolmas [2]. In particular, if the utility function is constructed from a recursive aggregator

and u (C) is homogeneous of degree γ, then the aggregator W (ct, ut+1) has the following

10

properties

W (λct,λγut+1) = λγW (ct, ut+1), (8)

Wc (λct,λγut+1) = λγ−1Wc (ct, ut+1) , (9)

Wu (λct,λγut+1) = Wu (ct, ut+1) . (10)

Moreover, the homogeneity of utility of degree γ implies that u(λct,λct+1, ...) = λγu(ct, ct+1, ...).

Since ut = u(ct, ct+1, ...), along any balanced growth path it follows that utilities in adjacent

periods are related by the expression, ut+1 = gγut.

Theorem 1 allows us to construct examples from homogenous aggregators for which we

can solve explicitly for the variables u, c, and k that solve the social planning problem along

a balanced growth path. To find such a solution, first, define the function φ (c) by the

expression

φ (c) =W (c, gγφ (c))

where φ (c) is the utility associated with the balanced growth consumption sequence ct = gtc.

Now define the marginal rate of substitution as follows;

β (C, g) =∂u/∂ct+1∂u/∂ct

=Wu (ct, ut+1)Wc (ct+1, ut+2)

Wc (ct, ut+1). (11)

Along a balanced growth path, using homogeneity properties (8-10) this expression simplifies

to the following expression,

β (C, g) = gγ−1Wu (c, gγφ (c)) .

To solve for the solution to the planning problem, along a balanced growth path, one

solves three equations in three unknowns: These are (1) the Euler equation; (2) the resource

constraint and (3) the recursive definition of utility. Beginning with the Euler equation, the

homogeneity properties (8-10) imply that

gγ−1Wu (c, gγφ (c)) = β (g) = [fK (k, 1)]

−1 . (12)

11

where k is the ratio of capital to the productivity trend. We have used the fact that

fK (kt+1, gt+1) is homogeneous of degree 0 to write it as fK (k, 1) .We define β (g) to be the

function β (C, g) evaluated along any balanced growth sequence C = {c, cg, cg2..}. Equation(12) determines k as a function of β (g). To find c, use the resource constraint

c + gk = f (k, 1) . (13)

Finally, u is defined by the expression,

u = φ (c) . (14)

We should note that the solution to Equations (12)—(14) does not necessarily solve the

social planning problem, since it ignores the initial condition (4). If the initial capital stock,

k happens to equal the value of k that solves the social planning optimum then the social

planning solution will be on the balanced growth path at all points in time. If the initial

condition is anything else, the social planning optimum may or may not converge to the

balanced growth path. A proof of local convergence require us to put more structure on

preferences.

5 An Economy with Two Agents

In this section we extend our analysis to an environment with two agents with preferences

represented by utility functions ui : X (g)→ R for i = {1, 2} each of which is defined by anaggregator

ui¡Ci¢=W i

¡πCi, ui(SCi)

¢. (15)

Consider the following social planning problem:

Problem 2

maxC1,C2

u = λu1¡C1¢+ (1− λ) u2

¡C2¢,

12

kt+1 = f¡kt, g

t¢− c1t − c2t t = 1, 2...

k0 = k,

where the technology f (k, gt) satisfies properties P1 − P6 and the utility functions ui (Ci),i = 1, 2, are generated by aggregators that satisfy assumptions U1− U6.

This is a concave programming problem and the necessary and sufficient conditions for

a solution are given by the equations

W 1c

¡c1t , u

1t+1

¢= W 1

u

¡c1t , u

1t+1

¢W 1c

¡c1t+1, u

1t+2

¢fK¡kt+1, g

t+1¢, t = 1, 2.. (16)

λW 1c

¡c1t , u

1t+1

¢= (1− λ)W 2

c

¡c2t , u

2t+1

¢, t = 1, 2.. (17)

W 1u

¡c1t , u

1t+1

¢= W 2

u

¡c2t , u

2t+1

¢, t = 1, 2.. (18)

limT→∞

¡λu1T

¡C1¢+ (1− λ)u2T

¡C2¢¢kT = 0, (19)

together with the initial condition

k0 = k. (20)

Since this economy is neoclassical, the first and second welfare theorems hold. It follows

that the equations that characterize the solution to Problem 2 are a subset of the equations

that define a competitive equilibrium and for any social welfare weight λ there will exists a

distribution of initial resources such that the social planning optimum can be decentralized

as a competitive equilibrium.

Definition 3 A balanced growth path for the two-person economy is a set of sequences

{K,C1, C2, Y } and a 4−tuple {k, c1, c2, y} such that

kt = kgt, c1t = c

1gt, c2t = c2gt yt = f

¡kt, g

t¢= ygt.

We will say that preferences and technologies are consistent with a balanced growth path

if there exist balanced growth sequences that satisfy the first order conditions for a planning

optimum (16—19).

13

We will be interested in heterogenous agent economies that are consistent with the ex-

istence of a balanced growth path. To be interesting economies, we will also require that

the income distribution is non-degenerate. In economies without growth, Lucas and Stokey

[7] have shown that the existence of an endogenous income distribution requires that the

agents’ discount rates must depend in a non-trivial way on their consumption sequences.

They define a condition (increasing marginal impatience) as the assumption that

W iuc (c,φ (c)) < 0,

where φ (c) = u (C) , and C is the constant sequence C = {c, c, ...} . In words, increasingmarginal impatience means that a representative agent, when faced with alternative constant

consumption sequences, will have a lower discount factor (smaller value of Wu (c,φ (c))) the

higher is his consumption path. As agents get richer, in this sense, they become more

impatient.

An analogous condition for a balanced growth economy would require that, faced with

two alternative balanced growth paths, C1 and C2 defined as

C1 = {c1, c1g, ...} ,

C2 = {c2, c2g, ...} ,

that

Wu (c1, gγφ (c1)) < Wu (c2, g

γφ (c2)) ,

whenever c1 > c2. This condition would require that as agents get richer, in the sense

of moving to strictly higher balanced growth paths, they become more impatient. But

Lemma 1 establishes that Wu (c, gγφ (c)) is independent of c. It follows that the assumption

of increasing marginal impatience is inconsistent with balanced growth.

The following theorem establishes that balanced growth economies require that all agents

be alike in a very strong sense.

14

Theorem 2 Consider a two-person economy with a technology that satisfies properties P1−P6. Let the preferences of the two agents be constructed from aggregator functions that satisfy

properties U1−U6. These preferences are consistent with balanced growth if and only if all,agents have the same constant rate of time preference, that is, if

W 1u

¡c1, gγφ

¡c1¢¢=W 2

u

¡c2, gγφ

¡c2¢¢= β (g) ,

where the functions W 1 (c2, gγφ (c1)) and W 2 (c1, gγφ (c2)), satisfy the homogeneity property

defined in Theorem 1.

Proof. In the one agent economy, consistency of preferences with balanced growth means

that the marginal rate of substitution must be constant and equal to the marginal rate of

transformation. For the two-person economy this condition must hold for each agent:

[fK (k, 1)]−1 = gγ−1W 1

u

¡c1, gγφ1

¡c1¢¢= gγ−1W 2

u

¡c2, gγφ2

¡c2¢¢. (21)

Lemma 1 implies that gγ−1W 1u

¡c1, gγφ1 (c1)

¢= β1 (g) and gγ−1W 2

u

¡c2, gγφ2 (c2)

¢= β2 (g)

are independent of the initial level of consumption along any two balanced growth paths

with the same growth factor g. It follows that a balanced growth equilibrium cannot exist

unless the discount factors of the two agents are equal.

The implication of theorem 2 is that the assumption of recursivity adds nothing of interest

to the study of the income distribution beyond the case of additively separable preferences.

Economies populated by agents with recursive homogeneous preferences like this do have

non-trivial wealth distributions, but these distributions are not endogenous in the sense of

Lucas-Stokey. Just as in the case of additively separable preferences, the distribution of

wealth in these economies is a function of initial conditions.

15

6 Time Dependent Preferences

In this section we suggest a way around this dilemma by introducing an exogenous time-

dependent factor into preferences. There are a number of interpretations to our analysis. In

Section 7 we pursue an example in which time enters as a result of a preference externality

because households care about their relative position in the income distribution.

We begin by expanding the commodity space. Consider aggregator functions W i :

X (g)2×R→R and let W i satisfy properties U1− U6 where W (x, y) refers to a function in

which x has two elements; the first represents consumption and the second represents an

exogenous sequence A = {a1, a2, a3...} where at = gt. The existence theorems of Boyd andDolmas do not restrict c to be a scalar, hence one can appeal to these theorems to assert

that there is a unique well defined solution ui(Ci, A) to the functional equation

ui(Ci, A) =W i¡πCi, πA, ui(SCi, SA)

¢, (22)

where πA = a1, SA = (a2, a3, ...). The social planner’s problem in the expanded economy

is defined in the same way as Problem 2 and since A is exogenous, the optimality conditions

given by (16—19) continue to hold.

The homogeneity properties (8—10) have the following analogs in the economy where

utility depends on time;

W (λct,λat,λγut+1) = λγW (ct, at, ut+1), (23)

Wc (λct,λat,λγut+1) = λγ−1Wc (ct, at, ut+1) , (24)

Wu (λct,λat,λγut+1) =Wu (ct, at, ut+1) . (25)

In order to find conditions under which there exists an equilibrium with a non-trivial

income distribution we need to define an analog to the property of increasing marginal

16

impatience that holds in an economy with growth. We proceed as follows. Define the

growth weighted variables

kt =ktgt, yt =

ytgt,

c1t =c1tgt, c2t =

c2tgt,

u1t =u1tgγt, u2t =

u2tgγt.

Now write the first order conditions for an optimum, in terms of these variables;

g1−γW 1c

¡c1t , 1, g

γ u1t+1¢= W 1

u

¡c1t , 1, g

γ u1t+1¢W 1c

¡c1t+1, 1, g

γu1t+2¢fK³kt+1, 1

´, t = 1, 2...

(26)

λW 1c

¡c1t , 1, g

γu1t+1¢= (1− λ)W 2

c

¡c2t , 1, g

γu2t+1¢

t = 1, 2... (27)

W 1u

¡c1t , 1, g

γu1t+1¢= W 2

u

¡c2t , 1, g

γu2t+1¢

t = 1, 2.... (28)

limT→∞

¡λu1T

¡C1, A

¢+ (1− λ)u2T

¡C2, A

¢¢kT = 0, (29)

and write the initial condition as

k0 = k. (30)

The variables ci and ui are constant along a balanced growth path and they are related

to each other by the functions φi (c) , i = 1, 2 where

φi¡ci¢=W i

¡ci, 1, gγφi

¡ci¢¢

is the utility of agent i, weighted by gγt, along the balanced growth path.

By assumption, u (C,A) is homogeneous in C and A. This assumption implies that the

functions W iu are homogenous of degree 0 in C and A; that is,

W iu

¡λci,λa,λγui

¢=W i

u

¡ci, a, ui

¢.

17

This does not imply zero degree homogeneity in C. The marginal rate of substitution along

a balanced growth path is described by the function;

β(ci, g) = gγ−1W iu

¡ci, 1,φi

¡ci¢¢,

which is not homogeneous of degree 0 in c. It follows the marginal rate of substitution

may be different along growth paths for which consumption grows at the same rate g. The

agent’s rate of time preference varies across different constant growth paths and is ordered

by the levels of the paths. By including the exogenous sequence A in preferences, we are

able to maintain consistency with balanced growth and allow discount factors to vary along

a balanced growth path.

7 Time Dependence and Balanced Growth

We now introduce an additional assumption that allows us to demonstrate the consistency

of balanced growth with an endogenous income distribution in the sense of Lucas and Stokey

in a two agent economy.

Definition 4 (Time Preference Variation) There exist numbers c1 and c2 such that

W 1u

¡c1, 1, gγφ

¡c1¢¢= W 2

u

¡c2, 1, gγφ

¡c2¢¢.

To demonstrate existence of an equilibrium, rates of time preference along the balanced

growth must vary sufficiently with consumption such that there exist consumption sequences

for which the discount rates of different agents are equated. That is the role of Definition 4.

The following theorem illustrates that if Definition 4 holds, there will exist a social planning

problem for which the solution is a balanced growth path.

Theorem 3 Consider a two-agent economy in which the technology satisfies condition P1−P6 and the preferences of each agent are constructed from two different aggregator functions

18

W i (c, a, u) , i = 1, 2, that satisfy assumptions U1 − U6. Assume further that preferences

satisfy the Definition 4 (time preference variation). Then there exists an initial value k0

and a welfare weight λ such that the solution to the Social Planning Problem 2 is a balanced

growth path {k, y, c1, c2}.

Proof. We first show that the first-order conditions (26—29) are consistent with the existence

of a balanced growth path. A balanced growth path is defined by the numbers {k, y, c1, c2}where kt = gtk, yt = gty, c1t = g

tc1, and c2t = gtc2. From the homogeneity of utility in C and

A, there exist numbers u1 and u2 such that u1t = gγtu1 and u2t = g

γtu2. Define the functions

φ1 (c1) and φ2 (c2) as follows,

φi¡ci¢= W i

¡ci, 1, gγφi

¡ci¢¢, i = 1, 2.

The first order condition (28), evaluated along a balanced growth path, requires that the

following equation should hold;

W 1u

¡c1, 1, gγφ1

¡c1¢¢=W 2

u

¡c2, 1, gγφi

¡c2¢¢. (31)

Let c1 and c2 be two different numbers that satisfy Equation (31). The existence of two

such numbers follows from Definition (4). Define the welfare weight λ by the expression

λW 1c

¡c1, 1, gγφ1

¡c1¢¢= (1− λ)W 2

c

¡c2, 1, gγφ1

¡c2¢¢, (32)

and define k to be the unique solution to the Equation

fk

³k, 1´=

1

W 1u

¡c1, 1, gγφ1 (c1)

¢ . (33)

The existence of a unique solution to this equation is implied by the Inada conditions,

(assumptions P5 and P6). We have demonstrated that our utility function is consistent

with the three first order conditions (16—18). Consistency with the transversality condition,

19

Equation (29) follows from U5 which bounds the discount factor and ensures that marginal

utilities grow more slowly than the growth rate of the economy.

Theorem 3 does not assert that the balanced growth path is consistent with equilibrium

for all welfare weights. For this to be true, one would require something analogous to the

Inada conditions applied to preferences. It asserts instead that if discount rates vary a little

bit with consumption then there exist welfare weights that are consistent with existence of a

balanced growth path. The implication for a decentralized equilibrium is that there exists

some initial wealth distribution that is consistent with the range of variation in discount

factors permitted by Definition 4, for which an equilibrium exists.

The theorem also says nothing about convergence to the balanced growth path. If the

initial capital stock happens to exactly equal the balanced growth stock k, then the solu-

tion to the social planner’s problem will place the economy immediately on its balanced

growth path. If the initial stock is not equal to k, the economy may or may not converge

to the balanced growth path. In the case of an exchange economy Lucas and Stokey give

a two-person example in which a non-inferiority condition plus the assumption of increasing

marginal impatience are sufficient to guarantee local convergence to balanced growth. Intu-

ition suggests that increasing marginal impatience is a necessary condition although we have

not been able to prove this for our model, nor have we been able to find sufficient conditions

that guarantee asymptotic stability of the balanced growth path.

8 Two Open Economy Examples

In this Section, we provide two examples that illustrate how our preferences may be used

in open economy models. We begin with an exogenous growth economy example and then

extend this example to an endogenous growth model in Subsection 8.2.

20

8.1 A Small Open Economy Example

Consider the example of a representative agent in a small open economy. Let world wealth

be given by the expression

at = gta

and let the individual’s preferences be constructed from the aggregator function

ut = W (ct, at, ut+1) ≡hcθta

ψ−θt + uδt+1a

ψ−δt

i 1ψ, (34)

where the aggregatorW satisfies assumptions U1−U6. We also assume, δ < 1, θ < 1 and

δ > ψ.2 Note that this aggregator is linearly homogenous.

This specification implies that as the world gets richer, this individual feels better off.

More importantly, as the world gets richer, his marginal utility of consumption increases

since he values his consumption sequence in part by comparing it to world wealth. In a rich

world an extra unit of consumption gives more additional utility than it would in a poor

world.

The individual faces the sequence of budget constraints

bt = bt−1R + wt − ct, t = 1, ...

b0 = b0,

where bt is domestic assets at date t, wt is exogenous wage income defined by the sequence

wt = gtw, and R is the exogenous market interest factor.

2Since the composition of concave functiuons is concave, concavity of the aggregator W (c, u0) in c and

u0 is a sufficient condition for u (C) to be concave in C. If δ and θ are strictly less than 1 then u (C) is

strictly concave when ψ/δ = 1. It follows from continuity that u (C) is strictly concave for ψ/δ in an open

neighborhood of ψ/δ = 1.

21

The utility maximizing individual will choose a consumption sequence that equates his

marginal rate of substitution to the world interest factor R;µctat

¶θ−1=

µct+1at+1

¶θ−1µut+1at+1

¶δ−ψδR

ψgδ−ψ. (35)

Equations (34) that defines recursive preferences and (35) that represents the first-order

condition for maximization of utility can be expressed as a system of two difference equations

in the transformed variables ut = ut/at and ct = ct/at;

uψt = cθt + gδuδt+1, (36)

cθ−1t =δR

ψgδ−ψ cθ−1t+1 u

δ−ψt+1 . (37)

Let bt be the ratio of net assets to world wealth, that is, bt = bt/at. The budget constraint,

in transformed variables, takes the form;

bt = bt−1R

g+ w − ct, t = 1, ... (38)

b0 = b0, (39)

where wt = wt/gt. Equations (36—38) represent a system of three difference equations in

three variables, ut, ct, bt with a single initial condition, b0. The unique balanced growth

path is defined by the equations

u =

µδR

ψ

¶ 1ψ−δ 1

g, c =

¡uψ − gδuδ¢ 1θ , b =

(c− w)R/g − 1 .

The Equations (36—37) form a sub-system in the two variables ut and ct that locally,

(around the balanced growth path) obeys the linear equations (1− s) δ 0

δ − ψ θ − 1

dut+1dct+1

= ψ −sθ0 θ − 1

dutdct

22

where dut+1 = (ut+1 − u) /u, dct+1 = (ct+1 − c) /c and s = cθ/uψ. This system has the

representation dut+1dct+1

= J dutdct

,where

J =

1(1−s)δψ

−1(1−s)δsθ

δ−ψ(1−s)δ(1−θ)ψ 1− δ−ψ

(1−s)δ(1−θ)sθ

.In the appendix we show that when δ > ψ this system has one root below 1 and one root

above 1. We also assume that R > g which is a necessary condition for the wealth of the

agent to be well defined. Under these assumptions, the subsystem (36—37) defines a linear

function

dut = µ1dct

and a scalar difference equation

dct+1 = µ2dct

where 0 < µ2 < 1. This subsystem converges back to the balanced growth path for ct in the

neighborhood of c. The initial value of this difference equation, c0 is chosen to satisfy the

transversality condition, that is, to ensure that

b0 +∞Xt=1

Qt0 (wt − ct) = 0,

where

Qt0 =1

(R/g)t−1.

Agents in our example display an endogenous discount rate and their long-run position in

the world income distribution will be a function of their preferences.

23

8.2 An Endogenous Growth Example

We now discuss an example of endogenous growth model in which agents have the same

preferences as those described in Subsection 8.1. We will show that this example gives

a novel interpretation of why two countries with different GDP per capita might converge

over time. In our example a country’s growth rate will increase or decrease because the

representative agent cares about his relative position in the world income distribution.

Let preferences be given by the same aggregator function as Subsection 8.1 but let the

technology be represented by the function

kt+1 = Rkt − ct, (40)

k0 = k0 (41)

As before, at = gta is world wealth. Since we assume that the same technology is avail-

able to all countries in the world there is no incentive to borrow and lend internationally;

the domestic interest factor R is equal to the world interest factor and is pinned down by

technology. The world growth rate g is a function of the preferences of the other countries

in the world.

Rewriting this system in transformed variables gives the system

uψt = cθt + gδuδt+1, (42)

cθ−1t =δR

ψgδ−ψ cθ−1t+1 u

δ−ψt+1 , (43)

kt+1 =R

gkt − ct

g, (44)

k0 = k (0) , (45)

which has a balanced growth path

u =

µδR

ψ

¶ 1ψ−δ 1

g, c =

¡uψ − gδuδ¢ 1θ , k =

c

R − g .

24

This economy has the same stability properties as the previous example, but a different

interpretation. Since R > g, Equation (44) is an unstable difference equation. Since the

subsystem (42—43) is a saddle, there is a unique value of u0 and a unique c0 such that the

system converges to the balanced growth path. During the transition, the economy may

grow faster than the world growth rate g, or it may grow slower.

If the economy starts out relatively poor, the domestic agent will devote a relatively

large share of capital to investment and, temporarily, the domestic economy will grow faster

than the world growth factor, g. As the economy becomes richer, the agent becomes more

impatient and domestic growth slows down until it reaches the world growth rate.

If the economy starts out relatively rich, the reverse happens. The agent consumes a

large fraction of his wealth and the economy grows slower than g. As the agent becomes

poorer he becomes more patient and the growth rate of the domestic economy increases.

All economies grow, asymptotically, with growth factor g because agents care about their

relative position in the world income distribution.

9 Conclusion

In this paper we have studied the circumstances under which one can model an endogenous

income distribution in a growing economy. We have shown that the homogeneity of the

welfare aggregator (a property that is required for balanced growth) has strong implications

in multiple agent environments. In general, balanced growth equilibria do not exist in a

multi-agent economy except for the special case where all agents have the same constant

rate of time preference. This case is uninteresting since it eliminates meaningful preference

heterogeneity which is one of the key motivations for studying recursive preferences to begin

with.

25

Given that balanced growth provides a fairly good description of the industrialized

economies, these findings highlight a problematic feature of recursive preferences. We have

suggested a generalization of the recursive preference structure that permits the coexistence

of balanced growth equilibria with multiple agent economies and recursive preferences. Our

extension requires preferences to be explicitly time dependent and the aggregator to be ho-

mogeneous in current consumption, future utility and an exogenous (time dependent) growth

factor. Our specification is consistent with models where the externality in preferences comes

through average per capita income or through per-capita world wealth.

26

AppendixThe Roots of J. The product of the roots (the determinant of determinant of J) is given

by

D =1

(1− s)ψ

δ

and the sum of the roots, (the Trace of J) is:

T =1

(1− s)ψ

δ+

1− ψ/δ

(1− s) (θ − 1)sθ + 1.

When ψ/δ = 1 the roots are given by 1 and 1/ (1− s) .The determinant is clearly increasing in ψ/δ. The trace is given by the expression

f (ψ/δ) =1

(1− s)ψ

δ+

1− ψ/δ

(1− s) (θ − 1)sθ + 1,

and since 0 < s < 1, the slope

∂f (ψ/δ)

∂ψ/δ=

1− θ (1− s)(1− s) (1− θ)

,

is positive for 0 < θ < 1 . Hence the trace is also increasing in ψ/δ.

Since the determinant and the trace are both increasing in ψ/δ it follows that as we

reduce ψ below δ, one root will remain above 1 and the other will decrease below 1. A proof

of this statement is providing by examining Figure 1.

The curve in Figure 1 plots the function

f (x) = x2 − TRACE x+DET.

The zeros of this function are the roots of the characteristic polynomial of J . The figure

shows f (x) for δ = ψ (the dashed curve) and f (x) for δ > ψ (the solid curve).

27

( )Root 1B δ ψ> >

( )Root 1B δ ψ= >

( )Root 1A δ ψ= =

( )Root 1A δ ψ> <

( )Trace δ ψ>

( )Trace δ ψ=

( )Det δ ψ>

( )Det δ ψ=

STABLE UNSTABLE

Figure 1

As ψ is lowered below δ, the trace (this is given by 2 argmin f (x)) and the determinant

(this is represented by the intercept) both decrease. If the two roots of the system before

and after the change are denoted {xA1, xB1} and {xA2, xB2}, the figure illustrates that thesetwo roots move from {xA1 = 1, xB1 > 1} to {xA2 < 1, xB2 > 1}. Hence, for values of ψ a

little below δ the subsystem defined by equations (36) and (37) is a saddle.

28

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